Ruslan Salimov

• National Academy of Sciences of Ukraine

Here we give a survey of consequences from the theory of the Beltrami equations in the complex plane ℂ to generalized Cauchy-Riemann equations ∇υ = B∇u in the real plane ℝ² and clarify the relationships of the latter to the A−harmonic equation div A grad u = 0 with the matrix-valued coefficients A, which is one of the main equations in the potentia...

Considering quasisymmetric mappings between b-metric spaces we have found a new estimation for the ratio of diameters of two subsets which are images of two bounded subsets. This result generalizes the well-known Tukia-V\"{a}is\"{a}l\"{a} inequality. The condition under which the image of a b-metric space under quasisymmetry is also a b-metric spac...

In 1980 P. Tukia and J. V\"{a}is\"{a}l\"{a} in seminal paper [P. Tukia and J. V\"ais\"al\"a, Quasisymmetric embeddings of metric spaces, Ann. Acad. Sci. Fenn., Ser. A I, Math. 5, 97--114 (1980)] extended a concept of quasisymmetric mapping known from the theory of quasiconformal mappings to the case of general metric spaces. They also found an esti...

In the present paper, we prove generalizations of Banach, Kannan, Chatterjea, \'Ciri\'c-Reich-Rus fixed point theorems, as well as of the fixed point theorem for mappings contracting perimeters of triangles. We consider corresponding mappings in semimetric spaces with triangle functions introduced by M. Bessenyei and Z. P\'ales. Such an approach al...

The class of quasisymmetric mappings on the real axis was first introduced by A. Beurling and L. V. Ahlfors in 1956. In 1980 P. Tukia and J. V\"{a}is\"{a}l\"{a} considered these mappings between general metric spaces. In our paper we generalize the concept of quasisymmetric mappings to the case of general semimetric spaces and study some properties...

We consider the class of ring $Q$-homeomorphisms with respect to $p$-modulus in $\mathbb{R}^{n}$ with $p > n$, and obtain lower bounds for limsups of the distance distortions under such mappings. These estimates can be treated as H\"{o}lder's continuity of the inverses near the origin. The sharpness is illustrated by example

In 2017 M. Bessenyei and Z. P\'ales introduced a definition of a triangle function which generates a concept of a generalized triangle inequality in semimetric spaces. Inspired by this concept we discuss already known inequalities in metric spaces that relate the six distances determined by four points and introduce a definition of a generalized fo...

In this paper, we consider refined geometric characterizations of weak p -quasiconformal mappings $$\varphi :\Omega \rightarrow \widetilde{\Omega }$$ φ : Ω → Ω ~ , where $$\Omega$$ Ω and $$\widetilde{\Omega }$$ Ω ~ are domains in $${\mathbb {R}}^n$$ R n . We prove that mappings with bounded geometric p -dilatation on the set $$\Omega {{\setminus }}...

Here we give a survey of consequences from the theory of the Beltrami equation in the complex plane $\mathbb C$ to generalized Cauchy-Riemann equation in the real plane $\mathbb R^2$ and clarify the relationships of the latter to the $A-$harmonic equation ${\rm div} A(z)\,{\rm grad}\, u(z) = 0$, $z=x+iy$, with matrix valued coefficients $A$ that is...

We investigate the asymptotic behavior at infinity of regular homeomorphic solutions of the nonlinear Beltrami equation with the Jacobian on the right-hand side. The sharpness of the above bounds is illustrated by several examples.

In the present study, we prove generalizations of Banach, Kannan, Chatterjea, Ćirić-Reich-Rus fixed point theorems, as well as of the fixed point theorem for mapping contracting perimeters of triangles. We consider corresponding mappings in semimetric spaces with triangle functions introduced by Bessenyei and Páles. Such an approach allows us to de...

This is a survey of our recent results concerning divergence-type linear and quasi-linear equations in the complex plane. It contains a number of existence, representation, and regularity theorems for the solutions of fundamental boundary value problems for such equations. The degeneration case of uniform ellipticity is also covered by means of the...

This is a survay of our recent results concerning divergence type linear and quasilinear equations in the complex plane. It contains a number of existence, representation and regularity theorems for solutions of fundamental boundary value problems for such equations. The degeneration case of uniform ellipticity is also covered by means of BMO, VMO,...

We prove a series of theorems on the asymptotic behavior of regular homeomorphic solutions of the nonlinear Cauchy–Riemann–Beltrami-type equation. Sufficient conditions for the logarithmic and exponential growth of regular solutions are obtained.

We study a nonlinear Beltrami equation $f_\theta =\sigma \,|f_r|^m f_r$ in polar coordinates $(r,\theta ),$ which becomes the classical Cauchy–Riemann system under $m=0$ and $\sigma =ir.$ Using the isoperimetric technique, various lower estimates for $|f(z)|/|z|, f(0)=0,$ as $z\to 0,$ are derived under appropriate integral conditions on complex/dir...

Here we give a survey of various integral criteria in terms of inner dilatations KI for the removability of isolated singularities of mappings with finite length distortion in ℝn, n ≥ 2, that are a natural extension of the well-known Martio–Väisäłä mappings with bounded length distortion. In particular, the survey includes many effective integral c...

We continue to investigate the regular homeomorphic solutions to nonlinear Beltrami equation introduced in Golberg and Salimov (Complex Var Elliptic Equ 65(1):6–21, 2020). Schwarz Lemma type estimates are obtained involving the length-area method. The lower bounds for the inverses are also established.

Here we give a survey of various integral criteria in terms of inner dilatations $K_{I}$ for the removability of isolated singularities of mappings with finite length distortion in $\mathbb{R}^{n}$, $n\geq 2$, that are a natural extension of the well-known Martio-Vaisal a mappings with bounded length distortion. In particular, the survey includes m...

The so-called ring Q-homeomorphisms with respect to p-modulus as p > 2 on the complex plane have been investigated. The lower bound for the distortion of the disk image diameter has been obtained. Extreme-value problems have been solved concerning the minimization of the distortion functional for the disk image diameter on some classes of ring Q-ho...

The so-called ring $Q$-homeomorphisms with respect to $p$-modulus as $p>2$ on the complex plane have been investigated. The lower bound for the distortion of the disk image diameter has been obtained. Extreme-value problems have been solved concerning the minimization of the distortion functional for the disk image diameter on some classes of ring...

The article is devoted to mappings with boundedand finite distortion of planar domains. Our investigations aredevoted to the connection between mappings of the Sobolev class andupper bounds for the distortion of the modulus of families of paths.For this class, we have proved the Poletsky-type inequality withrespect to the so-called inner dilatation...

In 2017, M. Bessenyei and Z. Páles [1] introduced a definition of a triangle function that generates a concept of a generalized triangle inequality in semimetric spaces. Inspired by this concept, we discuss already known inequalities in metric spaces that relate six distances determined by four points and introduce a definition of the generalized f...

УДК 517.5 Розглядається задача про локальну поведінку розв'язків рівнянь Бельтрамі в довільних областях. Знайдено достатні умови на комплексний коефіцієнт рівняння Бельтрамі, що забезпечують існування її розв'язку в довільній області, який є в ній неперервним за Гельдером. Ці результати можна застосовувати до крайових задач для рівняння Бельтрамі т...

We study the asymptotic behavior at infinity of ring Q-homeomorphisms with respect to p-modulus for p>n

In 2017, M. Bessenyei and Z. Páles [1] introduced a definition of a triangle function that generates a concept of a generalized triangle inequality in semimetric spaces. Inspired by this concept, we discuss already known inequalities in metric spaces that relate six distances determined by four points and introduce a definition of the generalized f...

УДК 517.54, 517.12 Досліджуються кільцеві Q -гомеоморфізми щодо p -модуля при p > 2 на комплексній площині. Отримано нижню оцінку спотворення трансфінітного діаметра образу круга. Розв'язано задачу про мінімізацію функціонала спотворення трасфінітного діаметра круга на деякому класі кільцевих Q -гомеоморфізмів щодо p -модуля.

In this paper we consider topological mappings defined by $p$-capacity inequalities in domains of $\mathbb R^n$. In the case $p=n-1$ we prove the Lipschitz continuity of such mappings, that extends the result by F.~W.~Gehring.

A sufficient condition providing a local finite Lipschitzness for the regular solutions of Sobolev class Wloc1,1 to a nonlinear Beltrami equation introduced in [Golberg A, Salimov R. Nonlinear Beltrami equation. Complex Var Elliptic Equ. 2020;65(1):6–21] is established. Some illustrating examples and preliminary results are also given.

УДК 517.5 Досліджується поведінка на нескінченності кільцевих Q -гомеоморфізмів щодо p -модуля при p > n .

In this article, we consider the H\"{o}lder continuity of injective maps in Orlicz-Sobolev classes defined on the unit ball. Under certain conditions on the growth of dilatations, we obtain the H\"{o}lder continuity of the indicated class of mappings. In particular, under certain special restrictions, we show that Lipschitz continuity of mappings h...

We continue to study regular homeomorphic solutions to the nonlinear Beltrami equation introduced in [24]. Estimates of the Schwarz Lemma type have been obtained using the length-area method.

We study the distortion features of homeomorphisms of Sobolev class W 1,1 loc admitting integrability for p-outer dilatation. We show that such map-pings belong to W 1,n−1 loc , are differentiable almost everywhere and possess absolute continuity in measure. In addition, such mappings are both ring and lower Q-homeomorphisms with appropriate measur...

The class of quasisymmetric mappings on the real axis was first introduced by Beurling and Ahlfors in 1956. In 1980 Tukia and Väisälä considered these mappings between general metric spaces. In our paper we generalize the concept of a quasisymmetric mapping to the case of general semimetric spaces and study some properties of these mappings. In par...

The article is devoted to mappings with bounded and finite distortion of plane domains. Our investigations are devoted to the connection between mappings of the Sobolev class and upper bounds for the distortion of the modulus of families of paths. For this class, we have proved the Poletsky-type inequality with respect to the so-called inner dilata...

In this paper, the estimate for growth of homeomorphic solutions of the Beltrami equation at infinity is obtained, provided that the dilatation quotient has a global finite mean oscillation.

In this article, we consider the H?lder continuity of injective maps in Orlicz-Sobolev classes defined on the unit ball. Under certain conditions on the growth of dilatations, we obtain the H?lder continuity of the indicated class of mappings. In particular, under certain special restrictions, we show that Lipschitz continuity of mappings holds. We...

We study regular solutions of the nonlinear Cauchy–Riemann–Beltrami equation for the logarithmic asymptotics in terms of the lower limits and solve an extreme problem for the functional disk image area in a certain class of solutions to the nonlinear Cauchy–Riemann–Beltrami system.

We study the behavior at infinity of ring Q-homeomorphisms with respect to p-modulus for p > n.KeywordsRing Q-homeomorphisms p-modulus of a family of curvesQuasiconformal mappingsCondenser p-capacity of a condenser Mathematics Subject Classification (2010) 30C65

Considering quasisymmetric mappings between b-metric spaces we have found a new estimation for the ratio of diameters of two subsets which are images of two bounded subsets. This result generalizes the well-known Tukia–Väisälä inequality. The condition under which the image of a b-metric space under a quasisymmetric mapping is also a b-metric space...

The study of nonlinear Cauchy--Riemann--Beltrami systems is conditioned study of certain problems of hydrodynamics and gas dynamics, in which there is an inhomogeneity of media and a certain singularity. The paper considers a nonlinear Cauchy--Riemann--Beltrami type system in the polar coordinate system in which the radial derivative is expressed t...

УДК 517.54; 517.12Дослiджуються регулярнi розв’язки нелiнiйної системи Кошi – Рiмана – Бельтрамi на логарифмiчну асимптотику у термiнах нижнiх границь. Розв’язано екстремальну задачу для функцiонала площi образу круга на деякому класi розв’язкiв нелiнiйної системи Кошi – Рiмана – Бельтрамi.

Considering quasisymmetric mappings between b-metric spaces we have found a new estimation for the ratio of diameters of two subsets which are images of two bounded subsets. This result generalizes the well-known Tukia-Vaisala inequality. The condition under which the image of a b-metric space under a quasisymmetric mapping is also a b-metric space...

We study the problem of continuation on boundary of the so-called Q-homeomorphisms between domains in metric spaces with measures. We obtain conditions for function Q(x) and domain boundary, under which any homeomorphism allows continuous continuation on boundary.

A power estimate of the area of the image of a disk for regular homeomorphisms possessing the Luzin N-property is obtained in terms of the p-angular dilation for p > 2. The result generalizes the known estimate by M.A. Lavrent’ev. A number of theorems on the asymptotic behavior of regular homeomorphic solutions of the nonlinear Beltrami equation ar...

We study the behavior at infinity of ring $Q$-homeomorphisms with respect to $p$-modulus for $p>n$.

We study homeomorphisms and mappings with branching in domains of the Euclidean space. We establish pointwise HÖlder and Lipschitz properties of mappings whose characteristics satisfy a Dini-type condition or whose mean values over infinitesimal balls are finite at the corresponding points. Moreover, we find conditions on the complex coefficients o...

A power estimate of the area of the image of a disk for regular homeomorphisms possessing the Luzin N-property is obtained in terms of the p-angular dilation for p>2. The result generalizes the known estimate by M.A. Lavrent'ev. A number of theorems on the asymptotic behavior of regular homeomorphic solutions of the nonlinear Beltrami equation are...

We consider ring Q-homeomorphisms with respect to the p-modulus in the space ℝⁿ for p>n. A lower bound for the volume of the image of a ball under these mappings is obtained. We solve the extreme problems of minimization of functionals of the volume of the image of a ball and the area of the image of a sphere.

Знайдено умови на зовнішню дилатацію KO(x, f ) та межі областей Rn, n ≥ 3, при яких гомеоморфізми f класів Соболєва W 1,1 loc допускають неперервне або гомеоморфне продовження в замикання областей. Ключові слова: класи Соболєва, критичний показник, зовнішня дилатація, гранична поведінка, неперервне і гомеоморфне провдовження.

We study the asymptotic behavior of the ratio |f(z)| / |z| as z→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z\rightarrow 0$$\end{document} for mappings differentia...

It is established that an arbitrary homeomorphism f in the Sobolev class W1,n−1loc with the outer dilatation K0(x,f)∈Ln−1loc is the socalled lower Q - homeomorphism with Q=K0(x,f) and the ring Q* homeomorphism with Q∗=Kn−10(x,f). These results make it possible to research the local and boundary behaviors of the mappings W1,n−1loc

We introduce a nonlinear counterpart of the classical Beltrami equation and study the main features of its solutions. It involves directional dilatations connected with a priory fixed point and a class of mappings called ring Q-homeomorphisms with respect to p-module. We also establish some regularity properties of solutions to such equation and il...

It is established that any homeomorphism f of the Sobolev class \( {W}_{\mathrm{loc}}^{1,1} \) with outer dilatation \( {K}_O\left(x,f\right)\in {L}_{\mathrm{loc}}^{n-1} \) is the so-called lower Q-homeomorphism with Q(x) = KO(x, f) and also a ring Q-homeomorphism with \( Q(x)={K}_O^{n-1}\left(x,f\right) \). This allows us to apply the theory of bo...

For a class of mappings satisfying upper modular estimates with respect to the families of curves, we study the local behavior of the corresponding inverse mappings. In terms of prime ends, we prove that the families of these homeomorphisms are equicontinuous (normal) in the closure of a given domain.

Досліджуються регулярні гомеоморфні розв’язки нелінійного рівняння Бельтрамі на степеневий і логарифмічний порядок росту. Побудовані розв’язки, що показують точність порядку росту в знайдених оцінках.

We study mappings with branching of a domain of Euclidean space. The H\"older and Lipschitz continuity are established for one class of spatial mappings whose characteristic satisfies the Dini type condition in a given domain. In addition, we found conditions on the complex coefficient of the degenerate Beltrami equations in the unit disk under whi...

In the present paper, it is found conditions on the complex coefficient of the Beltrami equations with the degeneration of the uniform ellipticity in the unit disk under which their generalized homeomorphic solutions are continuous by Hölder on the boundary. These results can be applied to the investigations of various boundary value problems for t...

The asymptotic behavior of lower Q-homeomorphisms relative to a p-modulus in ℝⁿ, n ≥ 2, at a point is studied. A number of logarithmic estimates for the lower limits under various conditions imposed on the function Q are obtained. Some applications of these results to the Orlicz–Sobolev classes \( {W}_{\mathrm{loc}}^{1,\varphi } \) in ℝⁿ, n ≥ 3 und...

We study metric properties of ring Q-homeomorphisms with respect to the p-modulus, p > 2, in the complex plane and establish lower bounds for the areas of disks. An extremal problem concerning minimization of the area functional is also solved.

We consider the class of ring Q-homeomorphisms with respect to p-modulus in ℝⁿ with p > n, and obtain a lower bound for the volume of images of a ball under such mappings. In particular, the following theorem is proved in the paper: Let D be a bounded domain in ℝⁿ, n ≥ 2 and let f: D → ℝⁿ be a ring Q-homeomorphism with respect to p-modulus at a poi...

We study the asymptotic behavior of the ratio $|f(z)|/|z|$ as $z\to 0$ for mappings differentiable a.e. in the unit disc with non-degenerated Jacobian. The main tools involve the length-area functionals and angular dilatations depending on some real number $p.$ The results are applied to homeomorphic solutions of a nonlinear Beltrami equation. The...

We study the asymptotic behavior of the ratio | f (z)|/|z| as z → 0 for mappings differentiable a.e. in the unit disc with non-degenerated Jacobian. The main tools involve the length-area functionals and angular dilatations depending on some real number p. The results are applied to homeomorphic solutions of a nonlinear Beltrami equation. The estim...

We study the asymptotic behavior of the ratio | f (z)|/|z| as z → 0 for mappings differentiable a.e. in the unit disc with non-degenerated Jacobian. The main tools involve the length-area functionals and angular dilatations depending on some real number p. The results are applied to homeomorphic solutions of a nonlinear Beltrami equation. The estim...

We consider certain classes of homeomorphisms of domains in \(\mathbb R^n\) with integrally bounded p-moduli of the families of curves and surfaces, which essentially extend the well-known classes of mappings such as quasiconformal, quasiisometric, Lipschitzian, etc. In the paper we survey the known results in this field regarded to the differentia...

We continue the study of homeomorphisms preserving integrally controlled weighted p-module of the ring domains. It was established earlier that under appropriate growth condition for the spherical mean of the weight such mappings are locally Hölder continuous with respect to logarithms of the distances. In this paper, we consider much more general...

We consider mappings satisfying one modular inequality with respect to cylinders in the space. The distortion of the modulus is majorized by an integral depending on a certain locally integrable function. We also prove a result on the absolute continuity of the analyzed mappings on lines. © 2017 Springer Science+Business Media, LLC, part of Springe...

We study open discrete mappings whose p-module of curve families is integrally restricted and establish various estimates for the Jacobian and dilatation coefficients. We also show that such mappings are close to Lipschitz mappings, quasiregular mappings and mappings of finite distortion. The sharpness of these estimates is illustrated by several e...

We study the local behavior of closed-open discrete mappings of the Orlicz–Sobolev classes in ℝⁿ; n ≥ 3: It is proved that the indicated mappings have continuous extensions to an isolated boundary point x0 of a domain D/{x0}, whenever its inner dilatation of order p ∈ (n − 1; n] has FMO (finite mean oscillation) at this point, and, in addition, the...

Получена оценка сверху для меры образа шара в некотором классе отображений, являющихся обобщением пространственных квазиизометрий с ветвлением. Как следствие, для указанных отображений получен аналог классической леммы Шварца при некотором дополнительном ограничении интегрального характера. Полученные результаты имеют соответствующие приложения в к...

There are established some sufficient conditions for boundary homeomorphic extension in metric spaces in which the measure of a ball of radius ε is controlled from above by a wide class of functions depending on ε. We consider a class of mappings whose ring moduli are integrally majorated. These results involve a finite mean oscillation and asympto...

An upper bound for the measure of the image of a ball under mappings of a certain class generalizing the class of branched spatial quasi-isometries is determined. As a corollary, an analog of Schwarz’ classical lemma for these mappings is proved under an additional constraint of integral character. The obtained results have applications to the clas...

In the work we consider Q-homeomorphisms w.r.t p-modulus on the complex plane as p > 2. We obtain a lower bound for the area of the image of a circle under such mappings. We solve the extremal problem on minimizing the functional of the area of the image of a circle.

The homeomorphisms of the Orlicz–Sobolev class Wloc1,φ under a condition of the Calderón type on φ in ℝn, n ≥ 3 are considered. For these classes of mappings, a number of theorems on the local behavior are established, and, in particular, an analog of the famous Gehring theorem on a local Lipschitz property, as well as various theorems on estimates...

A behavior of one class of mappings with finite distortion at a neighborhood of the origin is investigated. There is proved a lower estimate of distortion of a distance under mappings mentioned above.

We establish differentiability almost everywhere for homeomorphisms whose p-moduli are integrally controlled with weights containing the generic measurable functions.

Розглянуто кільцеві Q-гомеоморфізми відносно p-модуля на комплексній площині при p > 2. Для таких класів відоб-ражень встановлено оцінки знизу площі образу круга. Розв’язано екстремальну задачу про мінімізацію функціонала площі образу круга.

The paper is devoted to the development of the theory of lower Q-homeomorphisms relative to a p-modulus in ℝⁿ, n ≥ 2. For these classes of mappings, a number of theorems on the local behavior are established, and, in particular, an analog of the famous Gehring theorem on a local Lipschitz property is proved, various theorems on estimates of distort...

For a subtype of mappings with finite distortion f : D → D′, D, D′ ⊂ ℝⁿ; n ≥ 2; which admit the existence of branch points, a modular inequality playing an essential role in the study of various problems of planar and spatial mappings is established. As an application, the problem of removal of isolated singularities of open discrete mappings with...

We establish a series of new criteria of equicontinuity and, hence, normality of the mappings of Orlicz–Sobolev classes in terms of inner dilatations.

In this article we consider Q-homeomorphisms with respect to the p-modulus on the complex plane with p>2. It is obtained a lower area estimate for image of discs under such mappings. We solved the extremal problem about minimization of the area functional of images of discs.

For some class of mappings satisfying upper modular estimates with respect to families of curves, a behavior of the corresponding inverse mappings is investigated. In the terms of prime ends, it is proved that, families of such homeomorphisms are equicontinuous (normal) in the closure of a given domain.

For regular homeomorphisms of Sobolev class $W^{1,1}_{\textrm{loc}}$ having the Luzin $N$-property, it is established the estimation of the area of a disk image in terms of an angular dilatation. As a corollary, the analog of the well-known Ikoma-Schwartz lemma for such mappings is obtained.

It is found a new sufficient condition of finite Lipschitz in terms of inner dilation for homeomorphisms of the Orlicz-Sobolev class W1,φlocWloc1,φ under a condition of the Calderon type on φφ.

Q-гомеоморфизмов (Представлено членом-корреспондентом НАН Украины Ю. Ю. Трохимчуком) Рассмотрены нижние Q-гомеоморфизмы относительно p-модуля при p n. Для таких классов отображений установлена оценка сверху меры образа шара и, как следствие, получен аналог известной леммы Икома–Шварца. Приведенная оценка является далеко идущим обобщением хорошо изв...

The problem of extension of ring Q-homeomorphisms to the boundary between domains in λ(ε)-regular metric spaces is investigated. The conditions imposed on the function Q(x) and the boundaries of domains under which every ring Q-homeomorphism admits a continuous or homeomorphic extension to the boundary are formulated.

We consider the generic discrete open continuous mappings in ℝn under which the perturbation of extremal lengths of curve collections is controlled integrally via ∫ Q(x)η p (|x−x 0|)dm(x) with p ∈ (n−1, n), where Q is a measurable function on ℝn and \(\int_{{r_1}}^{{r_2}} {\eta (r)dr \geqslant 1} \) for any η on a given interval [r 1, r 2]. The mai...